randRange( -5, 5 ) randRange( -5, 5 ) randRange( -5, 5 ) -2 * A * H A * H * H + K ( function( x ){ return B * x + C; }) "initialized later"

a\cdot x^2 + b\cdot x + c is graphed below. Determine the signs of a, b, and c.

init({ range: [ [ -10, 10 ], [ -10, 10 ] ], scale: [ 25, 25 ] }); grid( [ -10, 10 ], [ -10, 10 ], { stroke: "#e2e2e2" }); // draw axes style({ stroke: "#000000", strokeWidth: 2 }); path( [ [ -10, 0 ], [ 10, 0 ] ] ); path( [ [ 0, -10 ], [ 0, 10 ] ] ); // graph style({ stroke: "#FF0000", strokeWidth: 2}); plot( function(x) { return A * x * x + B * x + C; }, [ -10,10 ] );

a is (A > 0) ? 'positive' : 'negative'

b is (B > 0) ? 'positive' : ( (B < 0) ? 'negative' : 'zero' )

c is (C > 0) ? 'positive' : ( (C < 0) ? 'negative' : 'zero' )

The number a determines how the legs are oriented. Is the parabola smiling or frowning?

A smiling parabola means a is positive and a frowning parabola means a is negative.

The parabola is ( A > 0) ? "smiling" : "frowning" thus a is ( A > 0 ) ? "positive" : "negative".

The number c determines where the parabola intersects the y-axis. Is the positive or negative part?

If the parabola intersected the positive part of the y-axis, then c would be positive.

The parabola intersects the y-axis in the point (0,c) = (0,C), thus c is ( C > 0 ) ? 'positive' : ( (C < 0) ? 'negative' : 'zero' ).

The number b determines how the parabola intersects the y-axis. Imagine the tangent at the intersection. What is the slope?

style({ stroke: "#FF8800", strokeWidth: 2}); line( [ -10, F( -10 )], [ 10, F( 10 )]);

The tangent where the parabola intersects the y-axis was drawn in orange. The number b is the slope.

The tangent has a (B > 0) ? 'positive' : ( (B < 0) ? 'negative' : 'zero' ) slope, so b is (B > 0) ? 'positive' : ( (B < 0) ? 'negative' : 'zero' ).

a is ( A > 0 ) ? "positive" : "negative", b is (B > 0) ? 'positive' : ( (B < 0) ? 'negative' : 'zero' ), and c is (C > 0) ? 'positive' : ( (C < 0) ? 'negative' : 'zero' ).